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<title>bitcoin_whitepaper_en</title><link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: "Lucida Console",Consolas,"Courier",monospace; }
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kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background-color: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; background-position: initial initial; background-repeat: initial initial; }
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code { text-align: left; }
a.md-print-anchor { white-space: pre !important; border: none !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; text-shadow: initial !important; background-position: 0px 0px !important; background-repeat: initial initial !important; }
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.MathJax_SVG .MJX-monospace { font-family: var(--monospace); }
.MathJax_SVG .MJX-sans-serif { font-family: sans-serif; }
.MathJax_SVG { display: inline; font-style: normal; font-weight: 400; line-height: normal; zoom: 90%; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; }
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table tr th { border-bottom-width: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }


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.CodeMirror { position: relative; overflow: hidden; }
.CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; background-color: inherit; background-position: inherit inherit; background-repeat: inherit inherit; }
.CodeMirror-sizer { position: relative; }
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.CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: hidden; }
.CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; }
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.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; border: none !important; background-position: 0px 0px !important; background-repeat: initial initial !important; }
.CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; }
.CodeMirror-lines { cursor: text; }
.CodeMirror pre { border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; border-width: 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; word-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; background-position: 0px 0px; background-repeat: initial initial; }
.CodeMirror-wrap pre { word-wrap: break-word; white-space: pre-wrap; word-break: normal; }
.CodeMirror-code pre { border-right-width: 30px; border-right-style: solid; border-right-color: transparent; width: fit-content; }
.CodeMirror-wrap .CodeMirror-code pre { border-right-style: none; width: auto; }
.CodeMirror-linebackground { position: absolute; left: 0px; right: 0px; top: 0px; bottom: 0px; z-index: 0; }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; }
.CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; }
.CodeMirror-measure pre { position: static; }
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.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; }
.cm-searching { background-color: rgba(255, 255, 0, 0.4); background-position: initial initial; background-repeat: initial initial; }
@media print { 
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:root {
    --side-bar-bg-color: #fafafa;
    --control-text-color: #777;
}

@include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);

html {
    font-size: 16px;
}

body {
    font-family: "Open Sans","Clear Sans","Helvetica Neue",Helvetica,Arial,sans-serif;
    color: rgb(51, 51, 51);
    line-height: 1.6;
}

#write {
    max-width: 860px;
  	margin: 0 auto;
  	padding: 30px;
    padding-bottom: 100px;
}
#write > ul:first-child,
#write > ol:first-child{
    margin-top: 30px;
}

a {
    color: #4183C4;
}
h1,
h2,
h3,
h4,
h5,
h6 {
    position: relative;
    margin-top: 1rem;
    margin-bottom: 1rem;
    font-weight: bold;
    line-height: 1.4;
    cursor: text;
}
h1:hover a.anchor,
h2:hover a.anchor,
h3:hover a.anchor,
h4:hover a.anchor,
h5:hover a.anchor,
h6:hover a.anchor {
    text-decoration: none;
}
h1 tt,
h1 code {
    font-size: inherit;
}
h2 tt,
h2 code {
    font-size: inherit;
}
h3 tt,
h3 code {
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}
h4 tt,
h4 code {
    font-size: inherit;
}
h5 tt,
h5 code {
    font-size: inherit;
}
h6 tt,
h6 code {
    font-size: inherit;
}
h1 {
    padding-bottom: .3em;
    font-size: 2.25em;
    line-height: 1.2;
    border-bottom: 1px solid #eee;
}
h2 {
   padding-bottom: .3em;
    font-size: 1.75em;
    line-height: 1.225;
    border-bottom: 1px solid #eee;
}
h3 {
    font-size: 1.5em;
    line-height: 1.43;
}
h4 {
    font-size: 1.25em;
}
h5 {
    font-size: 1em;
}
h6 {
   font-size: 1em;
    color: #777;
}
p,
blockquote,
ul,
ol,
dl,
table{
    margin: 0.8em 0;
}
li>ol,
li>ul {
    margin: 0 0;
}
hr {
    height: 2px;
    padding: 0;
    margin: 16px 0;
    background-color: #e7e7e7;
    border: 0 none;
    overflow: hidden;
    box-sizing: content-box;
}

li p.first {
    display: inline-block;
}
ul,
ol {
    padding-left: 30px;
}
ul:first-child,
ol:first-child {
    margin-top: 0;
}
ul:last-child,
ol:last-child {
    margin-bottom: 0;
}
blockquote {
    border-left: 4px solid #dfe2e5;
    padding: 0 15px;
    color: #777777;
}
blockquote blockquote {
    padding-right: 0;
}
table {
    padding: 0;
    word-break: initial;
}
table tr {
    border-top: 1px solid #dfe2e5;
    margin: 0;
    padding: 0;
}
table tr:nth-child(2n),
thead {
    background-color: #f8f8f8;
}
table tr th {
    font-weight: bold;
    border: 1px solid #dfe2e5;
    border-bottom: 0;
    margin: 0;
    padding: 6px 13px;
}
table tr td {
    border: 1px solid #dfe2e5;
    margin: 0;
    padding: 6px 13px;
}
table tr th:first-child,
table tr td:first-child {
    margin-top: 0;
}
table tr th:last-child,
table tr td:last-child {
    margin-bottom: 0;
}

.CodeMirror-lines {
    padding-left: 4px;
}

.code-tooltip {
    box-shadow: 0 1px 1px 0 rgba(0,28,36,.3);
    border-top: 1px solid #eef2f2;
}

.md-fences,
code,
tt {
    border: 1px solid #e7eaed;
    background-color: #f8f8f8;
    border-radius: 3px;
    padding: 0;
    padding: 2px 4px 0px 4px;
    font-size: 0.9em;
}

code {
    background-color: #f3f4f4;
    padding: 0 2px 0 2px;
}

.md-fences {
    margin-bottom: 15px;
    margin-top: 15px;
    padding-top: 8px;
    padding-bottom: 6px;
}


.md-task-list-item > input {
  margin-left: -1.3em;
}

@media print {
    html {
        font-size: 13px;
    }
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<div  id='write'  class = 'is-mac'><h1><a name="bitcoin-a-peer-to-peer-electronic-cash-system" class="md-header-anchor"></a><span>Bitcoin: A Peer-to-Peer Electronic Cash System</span></h1><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span>​</span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>作者：中本聪</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>satoshin@gmx.com</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>www.bitcoin.org </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>2008.10.31</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>中文翻译：李笑来</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>lixiaolai@gmail.com</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>2018.10.31</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span>[Checkout Github Repo for this translation](https://github.com/xiaolai/bitcoin-whitepaper-chinese-translation)</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 242px;"></div><div class="CodeMirror-gutters" style="display: none; height: 242px;"></div></div></div></pre><p>&nbsp;</p><blockquote><p><strong><span>Abstract.</span></strong><span> A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution. Digital signatures provide part of the solution, but the main benefits are lost if a trusted third party is still required to prevent double-spending. We propose a solution to the double-spending problem using a peer-to-peer network. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work, forming a record that cannot be changed without redoing the proof-of-work. The longest chain not only serves as proof of the sequence of events witnessed, but proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they&#39;ll generate the longest chain and outpace attackers. The network itself requires minimal structure. Messages are broadcast on a best effort basis, and nodes can leave and rejoin the network at will, accepting the longest proof-of-work chain as proof of what happened while they were gone. </span></p></blockquote><hr /><h2><a name="1-introduction" class="md-header-anchor"></a><span>1. Introduction</span></h2><p><span>Commerce on the Internet has come to rely almost exclusively on financial institutions serving as trusted third parties to process electronic payments. While the system works well enough for most transactions, it still suffers from the inherent weaknesses of the trust based model. Completely non-reversible transactions are not really possible, since financial institutions cannot avoid mediating disputes. The cost of mediation increases transaction costs, limiting the minimum practical transaction size and cutting off the possibility for small casual transactions, and there is a broader cost in the loss of ability to make non-reversible payments for non-reversible services. With the possibility of reversal, the need for trust spreads. Merchants must be wary of their customers, hassling them for more information than they would otherwise need. A certain percentage of fraud is accepted as unavoidable. These costs and payment uncertainties can be avoided in person by using physical currency, but no mechanism exists to make payments over a communications channel without a trusted party.</span></p><p><span>What is needed is an electronic payment system based on cryptographic proof instead of trust, allowing any two willing parties to transact directly with each other without the need for a trusted third party. Transactions that are computationally impractical to reverse would protect sellers from fraud, and routine escrow mechanisms could easily be implemented to protect buyers. In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions. The system is secure as long as honest nodes collectively control more CPU power than any cooperating group of attacker nodes.</span></p><h2><a name="2-transactions" class="md-header-anchor"></a><span>2. Transactions</span></h2><p><span>We define an electronic coin as a chain of digital signatures. Each owner transfers the coin to the next by digitally signing a hash of the previous transaction and the public key of the next owner and adding these to the end of the coin. A payee can verify the signatures to verify the chain of ownership.</span></p><p><img src='../pics/transactions.svg' alt='' referrerPolicy='no-referrer' /></p><p><span>The problem of course is the payee can&#39;t verify that one of the owners did not double-spend the coin. A common solution is to introduce a trusted central authority, or mint, that checks every transaction for double spending. After each transaction, the coin must be returned to the mint to issue a new coin, and only coins issued directly from the mint are trusted not to be double-spent. The problem with this solution is that the fate of the entire money system depends on the company running the mint, with every transaction having to go through them, just like a bank.</span></p><p><span>We need a way for the payee to know that the previous owners did not sign any earlier transactions. For our purposes, the earliest transaction is the one that counts, so we don&#39;t care about later attempts to double-spend. The only way to confirm the absence of a transaction is to be aware of all transactions. In the mint based model, the mint was aware of all transactions and decided which arrived first. To accomplish this without a trusted party, transactions must be publicly announced</span><sup class='md-footnote'><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup><span>, and we need a system for participants to agree on a single history of the order in which they were received. The payee needs proof that at the time of each transaction, the majority of nodes agreed it was the first received.</span></p><h2><a name="3-timestamp-server" class="md-header-anchor"></a><span>3. Timestamp Server</span></h2><p><span>The solution we propose begins with a timestamp server. A timestamp server works by taking a hash of a block of items to be timestamped and widely publishing the hash, such as in a newspaper or Usenet post</span><sup class='md-footnote'><a href='#dfref-footnote-2' name='ref-footnote-2'>2</a></sup><span> </span><sup class='md-footnote'><a href='#dfref-footnote-3' name='ref-footnote-3'>3</a></sup><span> </span><sup class='md-footnote'><a href='#dfref-footnote-4' name='ref-footnote-4'>4</a></sup><span> </span><sup class='md-footnote'><a href='#dfref-footnote-5' name='ref-footnote-5'>5</a></sup><span>. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it.</span></p><p><img src='../pics/timestamp-server.svg' alt='' referrerPolicy='no-referrer' /></p><h2><a name="4-proof-of-work" class="md-header-anchor"></a><span>4. Proof-of-Work</span></h2><p><span>To implement a distributed timestamp server on a peer-to-peer basis, we will need to use a proof-of-work system similar to Adam Back&#39;s Hashcash</span><sup class='md-footnote'><a href='#dfref-footnote-6' name='ref-footnote-6'>6</a></sup><span>, rather than newspaper or Usenet posts. The proof-of-work involves scanning for a value that when hashed, such as with SHA-256, the hash begins with a number of zero bits. The average work required is exponential in the number of zero bits required and can be verified by executing a single hash.</span></p><p><span>For our timestamp network, we implement the proof-of-work by incrementing a nonce in the block until a value is found that gives the block&#39;s hash the required zero bits. Once the CPU effort has been expended to make it satisfy the proof-of-work, the block cannot be changed without redoing the work. As later blocks are chained after it, the work to change the block would include redoing all the blocks after it.</span></p><p><img src='../pics/proof-of-work.svg' alt='' referrerPolicy='no-referrer' /></p><p><span>The proof-of-work also solves the problem of determining representation in majority decision making. If the majority were based on one-IP-address-one-vote, it could be subverted by anyone able to allocate many IPs. Proof-of-work is essentially one-CPU-one-vote. The majority decision is represented by the longest chain, which has the greatest proof-of-work effort invested in it. If a majority of CPU power is controlled by honest nodes, the honest chain will grow the fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of the block and all blocks after it and then catch up with and surpass the work of the honest nodes. We will show later that the probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added.</span></p><p><span>To compensate for increasing hardware speed and varying interest in running nodes over time, the proof-of-work difficulty is determined by a moving average targeting an average number of blocks per hour. If they&#39;re generated too fast, the difficulty increases.</span></p><h2><a name="5-network" class="md-header-anchor"></a><span>5. Network</span></h2><p><span>The steps to run the network are as follows:</span></p><blockquote><ol start='' ><li><span>New transactions are broadcast to all nodes.</span></li><li><span>Each node collects new transactions into a block.</span></li><li><span>Each node works on finding a difficult proof-of-work for its block.</span></li><li><span>When a node finds a proof-of-work, it broadcasts the block to all nodes.</span></li><li><span>Nodes accept the block only if all transactions in it are valid and not already spent.</span></li><li><span>Nodes express their acceptance of the block by working on creating the next block in the chain, using the hash of the accepted block as the previous hash.</span></li></ol></blockquote><p><span>Nodes always consider the longest chain to be the correct one and will keep working on extending it. If two nodes broadcast different versions of the next block simultaneously, some nodes may receive one or the other first. In that case, they work on the first one they received, but save the other branch in case it becomes longer. The tie will be broken when the next proof-of-work is found and one branch becomes longer; the nodes that were working on the other branch will then switch to the longer one.</span></p><p><span>New transaction broadcasts do not necessarily need to reach all nodes. As long as they reach many nodes, they will get into a block before long. Block broadcasts are also tolerant of dropped messages. If a node does not receive a block, it will request it when it receives the next block and realizes it missed one.</span></p><h2><a name="6-incentive" class="md-header-anchor"></a><span>6. Incentive</span></h2><p><span>By convention, the first transaction in a block is a special transaction that starts a new coin owned by the creator of the block. This adds an incentive for nodes to support the network, and provides a way to initially distribute coins into circulation, since there is no central authority to issue them. The steady addition of a constant of amount of new coins is analogous to gold miners expending resources to add gold to circulation. In our case, it is CPU time and electricity that is expended.</span></p><p><span>The incentive can also be funded with transaction fees. If the output value of a transaction is less than its input value, the difference is a transaction fee that is added to the incentive value of the block containing the transaction. Once a predetermined number of coins have entered circulation, the incentive can transition entirely to transaction fees and be completely inflation free.</span></p><p><span>The incentive may help encourage nodes to stay honest. If a greedy attacker is able to assemble more CPU power than all the honest nodes, he would have to choose between using it to defraud people by stealing back his payments, or using it to generate new coins. He ought to find it more profitable to play by the rules, such rules that favour him with more new coins than everyone else combined, than to undermine the system and the validity of his own wealth.</span></p><h2><a name="7-reclaiming-disk-space" class="md-header-anchor"></a><span>7. Reclaiming Disk Space</span></h2><p><span>Once the latest transaction in a coin is buried under enough blocks, the spent transactions before it can be discarded to save disk space. To facilitate this without breaking the block&#39;s hash, transactions are hashed in a Merkle Tree</span><sup class='md-footnote'><a href='#dfref-footnote-2-1' name='ref-footnote-2-1'>2</a></sup><sup class='md-footnote'><a href='#dfref-footnote-5-1' name='ref-footnote-5-1'>5</a></sup><sup class='md-footnote'><a href='#dfref-footnote-7' name='ref-footnote-7'>7</a></sup><span>, with only the root included in the block&#39;s hash. Old blocks can then be compacted by stubbing off branches of the tree. The interior hashes do not need to be stored.</span></p><p><img src='../pics/reclaiming-disk-space.svg' alt='' referrerPolicy='no-referrer' /></p><p><span>A block header with no transactions would be about 80 bytes. If we suppose blocks are generated every 10 minutes, 80 bytes * 6 * 24 * 365 = 4.2MB per year. With computer systems typically selling with 2GB of RAM as of 2008, and Moore&#39;s Law predicting current growth of 1.2GB per year, storage should not be a problem even if the block headers must be kept in memory.</span></p><h2><a name="8-simplified-payment-verification" class="md-header-anchor"></a><span>8. Simplified Payment Verification</span></h2><p><span>It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain, which he can get by querying network nodes until he&#39;s convinced he has the longest chain, and obtain the Merkle branch linking the transaction to the block it&#39;s timestamped in. He can&#39;t check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it, and blocks added after it further confirm the network has accepted it.</span></p><p><img src='../pics/simplified-payment-verification.svg' alt='' referrerPolicy='no-referrer' /></p><p><span>As such, the verification is reliable as long as honest nodes control the network, but is more vulnerable if the network is overpowered by an attacker. While network nodes can verify transactions for themselves, the simplified method can be fooled by an attacker&#39;s fabricated transactions for as long as the attacker can continue to overpower the network. One strategy to protect against this would be to accept alerts from network nodes when they detect an invalid block, prompting the user&#39;s software to download the full block and alerted transactions to confirm the inconsistency. Businesses that receive frequent payments will probably still want to run their own nodes for more independent security and quicker verification.</span></p><h2><a name="9-combining-and-splitting-value" class="md-header-anchor"></a><span>9. Combining and Splitting Value</span></h2><p><span>Although it would be possible to handle coins individually, it would be unwieldy to make a separate transaction for every cent in a transfer. To allow value to be split and combined, transactions contain multiple inputs and outputs. Normally there will be either a single input from a larger previous transaction or multiple inputs combining smaller amounts, and at most two outputs: one for the payment, and one returning the change, if any, back to the sender.</span></p><p>&nbsp;</p><p><img src='../pics/combining-splitting-value.svg' alt='' referrerPolicy='no-referrer' /></p><p><span>It should be noted that fan-out, where a transaction depends on several transactions, and those transactions depend on many more, is not a problem here. There is never the need to extract a complete standalone copy of a transaction&#39;s history.</span></p><h2><a name="10-privacy" class="md-header-anchor"></a><span>10. Privacy</span></h2><p><span>The traditional banking model achieves a level of privacy by limiting access to information to the parties involved and the trusted third party. The necessity to announce all transactions publicly precludes this method, but privacy can still be maintained by breaking the flow of information in another place: by keeping public keys anonymous. The public can see that someone is sending an amount to someone else, but without information linking the transaction to anyone. This is similar to the level of information released by stock exchanges, where the time and size of individual trades, the &quot;tape&quot;, is made public, but without telling who the parties were.</span></p><p><img src='../pics/privacy.svg' alt='' referrerPolicy='no-referrer' /></p><p><span>As an additional firewall, a new key pair should be used for each transaction to keep them from being linked to a common owner. Some linking is still unavoidable with multi-input transactions, which necessarily reveal that their inputs were owned by the same owner. The risk is that if the owner of a key is revealed, linking could reveal other transactions that belonged to the same owner.</span></p><h2><a name="11-calculations" class="md-header-anchor"></a><span>11. Calculations</span></h2><p><span>We consider the scenario of an attacker trying to generate an alternate chain faster than the honest chain. Even if this is accomplished, it does not throw the system open to arbitrary changes, such as creating value out of thin air or taking money that never belonged to the attacker. Nodes are not going to accept an invalid transaction as payment, and honest nodes will never accept a block containing them. An attacker can only try to change one of his own transactions to take back money he recently spent.</span></p><p><span>The race between the honest chain and an attacker chain can be characterized as a Binomial Random Walk. The success event is the honest chain being extended by one block, increasing its lead by +1, and the failure event is the attacker&#39;s chain being extended by one block, reducing the gap by -1.</span></p><p><span>The probability of an attacker catching up from a given deficit is analogous to a Gambler&#39;s Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as follows</span><sup class='md-footnote'><a href='#dfref-footnote-8' name='ref-footnote-8'>8</a></sup><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n67" cid="n67" mdtype="math_block">
			
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      \large p &=& \text{ probability an honest node finds the next block}\\
      \large q &=& \text{ probability the attacker finds the next block}\\
      \large q_z &=& \text{ probability the attacker will ever catch up from z blocks behind}
\end{eqnarray*}</script></div></div><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n68" cid="n68" mdtype="math_block">
			
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				1 & \textit{if}\; p \leq q\\
				(q/p)^z & \textit{if}\; p > q
				\end{Bmatrix}</script></div></div><p><span>Given our assumption that </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.425ex" height="1.994ex" viewBox="-39 -605.1 2335.6 858.4" role="img" focusable="false" style="vertical-align: -0.588ex; margin-left: -0.091ex;"><defs><path stroke-width="0" id="E6-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="0" id="E6-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E6-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E6-MJMATHI-70" x="0" y="0"></use><use xlink:href="#E6-MJMAIN-3E" x="780" y="0"></use><use xlink:href="#E6-MJMATHI-71" x="1836" y="0"></use></g></svg></span><script type="math/tex">p \gt q</script><span>, the probability drops exponentially as the number of blocks the attacker has to catch up with increases. With the odds against him, if he doesn&#39;t make a lucky lunge forward early on, his chances become vanishingly small as he falls further behind.</span></p><p><span>We now consider how long the recipient of a new transaction needs to wait before being sufficiently certain the sender can&#39;t change the transaction. We assume the sender is an attacker who wants to make the recipient believe he paid him for a while, then switch it to pay back to himself after some time has passed. The receiver will be alerted when that happens, but the sender hopes it will be too late.</span></p><p><span>The receiver generates a new key pair and gives the public key to the sender shortly before signing. This prevents the sender from preparing a chain of blocks ahead of time by working on it continuously until he is lucky enough to get far enough ahead, then executing the transaction at that moment. Once the transaction is sent, the dishonest sender starts working in secret on a parallel chain containing an alternate version of his transaction.</span></p><p><span>The recipient waits until the transaction has been added to a block and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.087ex" height="1.41ex" viewBox="0 -504.6 468 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E8-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E8-MJMATHI-7A" x="0" y="0"></use></g></svg></span><script type="math/tex">z</script><span> blocks have been linked after it. He doesn&#39;t know the exact amount of progress the attacker has made, but assuming the honest blocks took the average expected time per block, the attacker&#39;s potential progress will be a Poisson distribution with expected value:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n73" cid="n73" mdtype="math_block">
			
		<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.051ex" height="5.963ex" viewBox="0 -1409.3 3897.1 2567.2" role="img" focusable="false" style="vertical-align: -2.689ex; max-width: 100%;"><defs><path stroke-width="0" id="E3-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E3-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="0" id="E3-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E3-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use transform="scale(1.2)" xlink:href="#E3-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(1.2)" xlink:href="#E3-MJMAIN-3D" x="860" y="0"></use><use transform="scale(1.2)" xlink:href="#E3-MJMATHI-7A" x="1916" y="0"></use><g transform="translate(2861,0)"><g transform="translate(144,0)"><rect stroke="none" width="747" height="72" x="0" y="264"></rect><use transform="scale(1.2)" xlink:href="#E3-MJMATHI-71" x="81" y="676"></use><use transform="scale(1.2)" xlink:href="#E3-MJMATHI-70" x="60" y="-686"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\large \lambda = z \frac qp</script></div></div><p><span>To get the probability the attacker could still catch up now, we multiply the Poisson density for each amount of progress he could have made by the probability he could catch up from that point:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n75" cid="n75" mdtype="math_block">
			
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				\begin{Bmatrix}
				(q/p)^{(z-k)} & \textit{if}\;k\leq z\\
				1 & \textit{if} \; k > z
				\end{Bmatrix}</script></div></div><p><span>Rearranging to avoid summing the infinite tail of the distribution...</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n77" cid="n77" mdtype="math_block">
			
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transform="scale(0.849)" xlink:href="#E5-MJMATHI-6B" x="1635" y="0"></use><use transform="scale(0.849)" xlink:href="#E5-MJMAIN-29" x="2156" y="0"></use></g></g><use transform="scale(1.2)" xlink:href="#E5-MJSZ2-29" x="6460" y="-1"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">\large 1 - \sum_{k=0}^{z} \frac{\lambda^k e^{-\lambda}}{k!}
				\left ( 1-(q/p)^{(z-k)} \right )</script></div></div><p><span>Converting to C code...</span></p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="c" style="page-break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="c"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-meta">#include &lt;math.h&gt;</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable-3">double</span> <span class="cm-def">AttackerSuccessProbability</span>(<span class="cm-variable-3">double</span> <span class="cm-variable">q</span>, <span class="cm-variable-3">int</span> <span class="cm-variable">z</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">{</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable-3">double</span> <span class="cm-variable">p</span> <span class="cm-operator">=</span> <span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">q</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable-3">double</span> <span class="cm-variable">lambda</span> <span class="cm-operator">=</span> <span class="cm-variable">z</span> <span class="cm-operator">*</span> (<span class="cm-variable">q</span> <span class="cm-operator">/</span> <span class="cm-variable">p</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable-3">double</span> <span class="cm-variable">sum</span> <span class="cm-operator">=</span> <span class="cm-number">1.0</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable-3">int</span> <span class="cm-variable">i</span>, <span class="cm-variable">k</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-keyword">for</span> (<span class="cm-variable">k</span> <span class="cm-operator">=</span> <span class="cm-number">0</span>; <span class="cm-variable">k</span> <span class="cm-operator">&lt;=</span> <span class="cm-variable">z</span>; <span class="cm-variable">k</span><span class="cm-operator">++</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span>{</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable-3">double</span> <span class="cm-variable">poisson</span> <span class="cm-operator">=</span> <span class="cm-variable">exp</span>(<span class="cm-operator">-</span><span class="cm-variable">lambda</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-keyword">for</span> (<span class="cm-variable">i</span> <span class="cm-operator">=</span> <span class="cm-number">1</span>; <span class="cm-variable">i</span> <span class="cm-operator">&lt;=</span> <span class="cm-variable">k</span>; <span class="cm-variable">i</span><span class="cm-operator">++</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable">poisson</span> <span class="cm-operator">*=</span> <span class="cm-variable">lambda</span> <span class="cm-operator">/</span> <span class="cm-variable">i</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-variable">sum</span> <span class="cm-operator">-=</span> <span class="cm-variable">poisson</span> <span class="cm-operator">*</span> (<span class="cm-number">1</span> <span class="cm-operator">-</span> <span class="cm-variable">pow</span>(<span class="cm-variable">q</span> <span class="cm-operator">/</span> <span class="cm-variable">p</span>, <span class="cm-variable">z</span> <span class="cm-operator">-</span> <span class="cm-variable">k</span>));</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span>}</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">  </span><span class="cm-keyword">return</span> <span class="cm-variable">sum</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 352px;"></div><div class="CodeMirror-gutters" style="display: none; height: 352px;"></div></div></div></pre><p><span>Running some results, we can see the probability drop off exponentially with </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.087ex" height="1.41ex" viewBox="0 -504.6 468 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E8-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E8-MJMATHI-7A" x="0" y="0"></use></g></svg></span><script type="math/tex">z</script><span>.</span></p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="" style="page-break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.1</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=0 &nbsp;  P=1.0000000</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=1 &nbsp;  P=0.2045873</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=2 &nbsp;  P=0.0509779</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=3 &nbsp;  P=0.0131722</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=4 &nbsp;  P=0.0034552</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=5 &nbsp;  P=0.0009137</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=6 &nbsp;  P=0.0002428</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=7 &nbsp;  P=0.0000647</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=8 &nbsp;  P=0.0000173</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=9 &nbsp;  P=0.0000046</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=10 &nbsp; P=0.0000012</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.3</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=0 &nbsp;  P=1.0000000</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=5 &nbsp;  P=0.1773523</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=10 &nbsp; P=0.0416605</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=15 &nbsp; P=0.0101008</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=20 &nbsp; P=0.0024804</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=25 &nbsp; P=0.0006132</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=30 &nbsp; P=0.0001522</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=35 &nbsp; P=0.0000379</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=40 &nbsp; P=0.0000095</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=45 &nbsp; P=0.0000024</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; z=50 &nbsp; P=0.0000006</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 550px;"></div><div class="CodeMirror-gutters" style="display: none; height: 550px;"></div></div></div></pre><p><span>Solving for P less than 0.1%...</span></p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; P &lt; 0.001</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.10 &nbsp; z=5</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.15 &nbsp; z=8</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.20 &nbsp; z=11</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.25 &nbsp; z=15</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.30 &nbsp; z=24</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.35 &nbsp; z=41</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.40 &nbsp; z=89</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; q=0.45 &nbsp; z=340</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 198px;"></div><div class="CodeMirror-gutters" style="display: none; height: 198px;"></div></div></div></pre><h2><a name="12-conclusion" class="md-header-anchor"></a><span>12. Conclusion</span></h2><p><span>We have proposed a system for electronic transactions without relying on trust. We started with the usual framework of coins made from digital signatures, which provides strong control of ownership, but is incomplete without a way to prevent double-spending. To solve this, we proposed a peer-to-peer network using proof-of-work to record a public history of transactions that quickly becomes computationally impractical for an attacker to change if honest nodes control a majority of CPU power. The network is robust in its unstructured simplicity. Nodes work all at once with little coordination. They do not need to be identified, since messages are not routed to any particular place and only need to be delivered on a best effort basis. Nodes can leave and rejoin the network at will, accepting the proof-of-work chain as proof of what happened while they were gone. They vote with their CPU power, expressing their acceptance of valid blocks by working on extending them and rejecting invalid blocks by refusing to work on them. Any needed rules and incentives can be enforced with this consensus mechanism.</span></p><hr /><h2><a name="references" class="md-header-anchor"></a><span>References</span></h2><div class='footnotes-area'  ><hr/>
<div class='footnote-line'><span class='md-fn-count'>1</span> <strong><span>b-money</span></strong><span> Dai Wei (1998-11-01) </span><a href='http://www.weidai.com/bmoney.txt' target='_blank' class='url'>http://www.weidai.com/bmoney.txt</a><a name='dfref-footnote-1' href='#ref-footnote-1' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>2</span> <strong><span>Design of a secure timestamping service with minimal trust requirements</span></strong><span> Henri Massias, Xavier Serret-Avila, Jean-Jacques Quisquater </span><em><span>20th Symposium on Information Theory in the Benelux</span></em><span> (1999-05) </span><a href='http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.6228' target='_blank' class='url'>http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.6228</a><a name='dfref-footnote-2' href='#ref-footnote-2' title='回到文档' class='reversefootnote' >↩</a><a name='dfref-footnote-2-1' href='#ref-footnote-2-1' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>3</span> <strong><span>How to time-stamp a digital document</span></strong><span> Stuart Haber, W.Scott Stornetta </span><em><span>Journal of Cryptology</span></em><span> (1991) </span><a href='https://doi.org/cwwxd4' target='_blank' class='url'>https://doi.org/cwwxd4</a><span> DOI: </span><a href='https://doi.org/10.1007/bf00196791'><span>10.1007/bf00196791</span></a><a name='dfref-footnote-3' href='#ref-footnote-3' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>4</span> <strong><span>Improving the Efficiency and Reliability of Digital Time-Stamping</span></strong><span> Dave Bayer, Stuart Haber, W. Scott Stornetta </span><em><span>Sequences II</span></em><span> (1993) </span><a href='https://doi.org/bn4rpx' target='_blank' class='url'>https://doi.org/bn4rpx</a><span> DOI: </span><a href='https://doi.org/10.1007/978-1-4613-9323-8_24'><span>10.1007/978-1-4613-9323-8_24</span></a><a name='dfref-footnote-4' href='#ref-footnote-4' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>5</span> <strong><span>Secure names for bit-strings</span></strong><span> Stuart Haber, W. Scott Stornetta </span><em><span>Proceedings of the 4th ACM conference on Computer and communications security - CCS ’97</span></em><span>(1997) </span><a href='https://doi.org/dtnrf6' target='_blank' class='url'>https://doi.org/dtnrf6</a><span> DOI: </span><a href='https://doi.org/10.1145/266420.266430'><span>10.1145/266420.266430</span></a><a name='dfref-footnote-5' href='#ref-footnote-5' title='回到文档' class='reversefootnote' >↩</a><a name='dfref-footnote-5-1' href='#ref-footnote-5-1' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>6</span> <strong><span>Hashcash - A Denial of Service Counter-Measure</span></strong><span> Adam Back (2002-08-01) </span><a href='http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.8' target='_blank' class='url'>http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.8</a><a name='dfref-footnote-6' href='#ref-footnote-6' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>7</span> <strong><span>Protocols for Public Key Cryptosystems</span></strong><span> Ralph C. Merkle </span><em><span>1980 IEEE Symposium on Security and Privacy</span></em><span> (1980-04) </span><a href='https://doi.org/bmvbd6' target='_blank' class='url'>https://doi.org/bmvbd6</a><span> DOI: </span><a href='https://doi.org/10.1109/sp.1980.10006'><span>10.1109/sp.1980.10006</span></a><a name='dfref-footnote-7' href='#ref-footnote-7' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>8</span> <strong><span>An Introduction to Probability Theory and its Applications</span></strong><span> William Feller </span><em><span>John Wiley &amp; Sons</span></em><span> (1957) </span><a href='https://archive.org/details/AnIntroductionToProbabilityTheoryAndItsApplicationsVolume1' target='_blank' class='url'>https://archive.org/details/AnIntroductionToProbabilityTheoryAndItsApplicationsVolume1</a><a name='dfref-footnote-8' href='#ref-footnote-8' title='回到文档' class='reversefootnote' >↩</a></div></div></div>
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